theorem 16.35 System of Euler–Lagrange equations

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:982 · p. 642

Rests on

depends_on lemma 16.20 Mixed form ¶
- depends_on corollary 7.44 Substitution and integration by parts ¶
  - depends_on proposition 7.31 Chain rule ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-15 ¶
  - depends_on proposition 7.30 Leibniz rule ¶
    - depends_on definition 7.12 Product of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-14 ¶
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - depends_on definition 7.41 Antiderivative ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-24 ¶
    - proves proof ch:05-real-analysis@proof-25 ¶
  - proves proof ch:05-real-analysis@proof-26 ¶
- depends_on lemma 16.19 du Bois-Reymond ¶
  - depends_on equation 16.18 eq:calcvar-dbr-hyp ¶
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
  - proves proof ch:14-calculus-of-variations@proof-9 ¶
- proves proof ch:14-calculus-of-variations@proof-10 ¶
depends_on theorem 16.22 Euler–Lagrange ¶
- depends_on equation 16.15 eq:calcvar-first-variation ¶
- depends_on lemma 16.20 Mixed form ¶ ↺
- depends_on proposition 16.16 Stationarity is necessary ¶
  - depends_on definition 16.15 Variation; the first variation ¶
    - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
    - depends_on equation 7.177 eq:ana-functional-derivative ¶
  - depends_on definition 16.13 Weak and strong extrema ¶
    - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
      - depends_on definition 16.11 Admissible class; functional ¶ ↺
      - depends_on definition 6.24 Metric ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - proves proof ch:14-calculus-of-variations@proof-6 ¶
- proves proof ch:14-calculus-of-variations@proof-11 ¶
proves proof ch:14-calculus-of-variations@proof-19 ¶

Supports

depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
- depends_on proposition 21.56 The two prescriptions and their difference ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Mixed form	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:993
`depends_on`	→	Euler–Lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:993
`depends_on`	←	The isoperimetric extremal is a circle	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:225
`depends_on`	←	Multiplier rule for pointwise constraints	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1208
`proves`	←	ch:14-calculus-of-variations@proof-19	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:996